Sen. Marco Rubio, R-Fla., and Rep. Trey Radel, R-Fla., will introduce legislation next week to delay the so-called individual mandate until six months after the Government Accountability Office says the sites are fully functional. The individual mandate, beginning in 2014, will require many people to be insured or pay a penalty under the Affordable Care Act.

"It is not fair that next year Americans will be punished for not buying ObamaCare when the website they are supposed to buy it on doesn't work," Rubio said in a statement Monday.

Sen. Lamar Alexander, R-Tenn., announced his intention to introduce legislation that would require the administration to produce weekly reports about the health care exchanges. The reports would detail everything from the number of enrollees in the exchanges to how the administration is fixing the website's problems.

"This bill will require the administration to be honest and transparent with the public, governors overseeing state exchanges, state consumer protection regulators, and decision makers in Congress," Alexander said in a statement Tuesday. "No more hiding the damage of the train wreck. Americans are on this train."

Republicans have also raised concerns about how much it will cost for the administration to resolve the issues with the website.

"To try and clean up the mess, President Obama is calling up a tech surge to overhaul the website, and sending the bill to the taxpayers," Sen. John Cornyn, R-Texas, said. "The taxpayers have already shelled out hundreds of millions of dollars to build the original website. How much more money is President Obama going to demand?"

After mounting criticism about the website's flaws, President Obama vowed Monday that the problems will be fixed.

"The website that's supposed to make it easy to apply for and purchase the insurance is not working the way it should for everybody," the president said Monday. "There's no sugarcoating it. The website has been too slow. People have getting stuck during the application process."

"There's no excuse for the problems," he said. "And these problems are getting fixed."